From here, start walking either north or south. Do not use talismans, Skip spells, or other methods of transport other than the basic movement arrows--if you do, you get the message:

You abandon this circumnavigation.

You will occasionally get a message along the way about your journey; for example, on the Beach beyond the Great Desert (if going south), you get:

You have a mighty powerful urge just to head South forever.

At 250S you get

You go South.
You feel so far from home, and your trip is only half done!
Keep heading Sorth!

(Yes this is exactly how you spell south if you are a GL.)

You get this same message heading north, despite that 250S is 75% complete. - Heyus2 04:54, 27 July 2008 (BST)

I passed this point using the 'Repeat action' option on the new interface (speedboating south) and forgot to check for this message. Now that I think about it, I don't remember seeing the 'head south forever' message when I hit the beach south of the Great Desert, so now I'm no longer certain I'm still officially on a circumnavigation (even though I'm sure I didn't use a talisman, taxi, or any other disqualifier). Is there something in your info.xml that you can look at to see if you're officially circumnavigating? - parking_god 16:43, 7 October 2008 (BST)

Even if you are still circumnavigating, you can't complete the quest unless you get that message--Solune 21:30, 7 October 2008 (BST)

At 110N (going south) or 48S (going north) you get

You have come so far! Better report to the 'King!

Circumnavigate the entire world of Cities. The world wraps around in the North-South direction; you can step from 600S to 195N. Once you've gone the entire way around, go to the Throne Room in the 'king's palace (not coincidentally located at latitude 14N) to gain your reward--most notably 100 Glories.
Update:

<darkling> They're what used to be called Victory Points.

So that's something.

Repeatability

<darkling> Yes, it can be repeated as often as you like.
<darkling> That's by design.

It seems the quest is intended to be completed over and over again. Good to know.

Tips:

Get a Sea Horsie. 2AP to move on ocean terrain, and they don't wear out like boats do.

Get a Winged Horsie. 1AP to move on ocean and most other terrain. 20AP to take off and land, though. You also get the ability to fly over most monsters.

Undergo musical training to control the winds, making sailing cost 1 AP with a Speedboat or Trawler (as long as you only go in one direction).

If you're heading north at around 80-86E, when you first reach the Frozen South, be careful you don't get stuck in Sniffleheim - you can tell if you have by the lack of coordinates and the giant mutant monsters.

Transitions from ocean to icy terrain can be difficult. Winged boots can bring you from ocean onto ice floe cheaply, but not the reverse(??). Launching into the northern ocean from a beach is much cheaper, during periods when the northern ocean is not frozen.